Berger inequality
From Encyclopedia of Mathematics
For a compact Riemannian manifold , let
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where is the ball around
with radius
, be the injectivity radius, and set
. Then the inequality
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holds, with equality if and only if is isometric to the standard sphere with diameter
.
This inequality relies on the Kazdan inequality applied to the Jacobi equation for operators
on
for a unit vector
. Here,
is the curvature operator,
is the parallel transport along the geodesic ray
, and
is the parallel translated curvature operator on
.
References
[a1] | M. Berger, "Une borne inférieure pour le volume d'une variété riemannienes en fonction du rayon d'injectivité" Ann. Inst. Fourier (Grenoble) , 30 (1980) pp. 259–265 |
[a2] | I. Chavel, "Riemannian geometry: A modern introduction" , Cambridge Univ. Press (1995) |
How to Cite This Entry:
Berger inequality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Berger_inequality&oldid=50215
Berger inequality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Berger_inequality&oldid=50215
This article was adapted from an original article by H. Kaul (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article